To rationalize the emergence of spatial localized deformations in terms of specific combinations of the underlying physical parameters, we set out to solve the continuous equations, (REF ), analytically in the limit of low shear modulus \(G\)  by employing weakly nonlinear analysis [1]}. We construct this solution on the half-axis \(0\le x\le L/2\)  by employing the mirror-symmetry of the vertical displacement around the midpoint \(x=0\)  (see SI Sec. S3).
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